Degree

Doctor of Philosophy (PhD)

Department

Division of Computer Science and Engineering

Document Type

Dissertation

Abstract

Markov chain Monte Carlo (MCMC) methods are simulations that explore complex statistical distributions, while bypassing the cumbersome requirement of a specific analytical expression for the target. This stochastic exploration of an uncertain parameter space comes at the expense of a large number of ``burn-in'' samples, and the computational complexity leads to the curse of dimensionality. Although at the exploration level, some methods have been proposed to accelerate the convergence of the algorithm, such as tempering, Hamiltonian Monte Carlo, Rao-redwellization, and scalable methods for better performance, they cannot avoid the stochastic nature of this exploration. We develop algorithms for the energy minimization of Riesz kernels. This problem arises in different fields, most notably in the construction of space-filling sequences of points where the singularity of kernels guarantees a strong repelling property between these points. We propose a generalized energy metric, where points are deterministically generated through pairwise interactions. We study the properties of these points, called Riesz particles. They inherit the properties of both a well-separated distance and a bounded covering radius. We embed them into the sequential Monte Carlo method, and build a novel sampler with a higher acceptance ratio and less time consumption than the state-of-art for Bayesian analysis. We verified this using experiments for parameter inference in a linear Gaussian state-space model with synthetic data and in a non-linear stochastic volatility model with real-world data.

In sequential Monte Carlo methods, a resampling strategy is employed to replace low-weight particles with those of higher weight that better represent the target distribution. The goal is to reduce the variance among particle weights, which in turn concentrates the distribution of effective particles. This concentration facilitates a more rapid and precise approximation of the hidden Markov model, particularly for the nonlinear case. Typically, the distribution of these particles is skewed. We introduce a method of repeated ergodicity within a deterministic domain, which utilizes the median for resampling. This approach has resulted in the lowest variance when compared to alternative resampling techniques. With the deterministic domain size being much smaller than the population size. Under reasonable assumptions regarding particle size, our algorithm outperforms contemporary methods. This has been substantiated through theoretical analysis and empirical testing on hidden Markov models, encompassing both linear and nonlinear cases.

We demonstrate our Riesz particle and variance reduction algorithms by applying them to the problem of finding the optimal camera placement for a motion capture system. For greater efficiency, the camera configuration is motivated by the need to achieve 3D realistic and dynamic effects. We convert each sensing requirement into the geometrical and optical constraints on sensor location. We develop a binary integer programming model with an occlusion culling factor, from which the 3D region of viewpoints that satisfies that constraint is computed by greedy heuristics with Riesz-particle scale optimization. The optimal camera configuration problem is $\mathcal{NP}$-hard. We prove that the performance ratio $H(k)$ for our model grows at most logarithmically, under mild assumptions.

Date

7-16-2024

Committee Chair

Baumgartner, Gerald

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